To expand on T's answer, convergence acceleration methods assume that the error terms of the sequence you are interested in are of a certain form.
One of the general-purpose methods for accelerating the convergence of a sequence is the Shanks transformation. For the case of a sequence of partial sums of a series, the Shanks transformation amounts to constructing a sequence of Padé approximants, which hopefully converge faster to the limit of the sequence of partial sums. (The justly famous Wynn ε algorithm is an efficient realization of the Shanks transformation)
Another popular technique is Richardson extrapolation, which assumes that the error can be expressed as a power series of some form. This is in effect the application of the usual algorithms for polynomial interpolation to estimate the limit of a sequence. (Richardson extrapolation is the machinery behind the Romberg algorithm for accelerating the convergence of the trapezoidal rule, since the error of the trapezoidal rule is expressible as a series in powers of the panel size.)
I have been deliberately vague here since you have given absolutely no information on the nature of your sequence. There are many convergence acceleration methods to choose from (a good reference is Brezinski and Redivo-Zaglia's Extrapolation Methods: Theory and Practice), and the best sequence transformation to use depends a lot on the provenance of your sequence.